Multiscale Finite Element approach for “weakly” random problems and related issues

DOI:10.1051/m2an/2013122 www.esaim-m2an.org

MULTISCALE FINITE ELEMENT APPROACH

FOR “WEAKLY” RANDOM PROBLEMS AND RELATED ISSUES

Claude Le Bris

, Fr´ ed´ eric Legoll

and Florian Thomines

Abstract.We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale Finite Element Method. Using numerical experiments, we demonstrate the efficiency of our approach and the computational speed-up with respect to a more standard approach. In the stationary setting, we provide a complete analysis of the approach, extending that available for the deterministic periodic setting.

Mathematics Subject Classification. 35B27, 65M60, 65M12, 35R60, 60H.

Received November 7, 2011.

Published online April 8, 2014.

1. Overview of our approach and results

The Multiscale Finite Element Method (henceforth abbreviated as MsFEM) is a popular numerical approach for multiscale problems (see [3,11,18,20,30–32,37–39]). It consists in a Galerkin approximation of the original problem over a finite dimensional space generated by basis functions that are specificallyadaptedto the problem under consideration.

This approach is popular for a twofold reason. First, its use is not restricted to multiscale problems that converge to a homogenized problem in the limit of vanishing ratio between the small scale and the macroscopic scale. It may be applied to much more general situations. Second, when the problem does converge to a homog- enization problem, the MsFEM approach is meant to approximate the solution of the problem with the small scaleεat its actual small value and not “only” in the asymptotic regimeε→0, which is the regime addressed by homogenization theory.

Keywords and phrases.Weakly stochastic homogenization, finite elements, Galerkin methods, highly oscillatory PDE.

1 CERMICS, ´Ecole Nationale des Ponts et Chauss´ees, Universit´e Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vall´ee Cedex 2, France.lebris@cermics.enpc.fr

2 INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France.

3 Laboratoire Navier, ´Ecole Nationale des Ponts et Chauss´ees, Universit´e Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne- La-Vall´ee Cedex 2, France.legoll@lami.enpc.fr;thominef@lami.enpc.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2014

ET AL.

To fix the ideas, consider the problem of finding u^εsolving

−div [A^ε∇u^ε] =f inD, u^ε= 0 on∂D, (1.1) on a bounded domain D ⊂R^d, withf ∈L²(D), and whereA^ε is a uniformly bounded, coercive matrix that varies at scaleε. A standard Finite Element Method (FEM) would require a space discretization of the domain at the scaleεin order to capture the oscillations ofu^εat scaleε. This is prohibitively expensive. The MsFEM aims at accurately approximating u^ε using a limited number of degrees of freedom. It does not require the matrixA^εto be periodic (namelyA^ε(x) =A_per(x/ε) for a fixed periodic matrix A_per) or stationary.

We now briefly describe the approach and present the aim of this article. Starting from a coarse mesh Th

with a standard (sayP1) Finite Element basis set of functions φ⁰_iL

i=1, generating the associated space Vh:= span(φ⁰_i, i= 1, . . . , L),

we first numerically build the MsFEM basis functionsφ^ε_i. Several definitions of these basis functions have been proposed in the literature (yielding different numerical methods), and we detail this in the sequel (seee.g.(2.7)- (2.8)-(2.9)). For the moment, it is sufficient to know that, to eachφ⁰_i, which varies at the macroscopic scale, is associated a function φ^ε_i, with variations at the scale ε. In practice, φ^ε_i is numerically computed (in fact, pre-computed), using the specificities of the problem addressed. These highly oscillatory functions φ^ε_i generate the finite dimensional space

Wh:= span(φ^ε_i, i= 1, . . . , L).

Note thatWh andVh share the same dimension.

We next define the MsFEM solution u_M using a Galerkin approximation of (1.1) on Wh, instead of Vh. Again, details will be given below. The MsFEM solutionu_M provided by the approach reads

u_M(x) = L i=1

(U_M)_i φ^ε_i(x),

for some coefficients{(U_M)_i}^L_i=1. Of course, these coefficients depend onε, but this dependency is kept implicit in the sequel.

We now turn our attention to the stochastic problem

−div [A^ε(·, ω)∇u^ε(·, ω)] =f inD, u^ε(·, ω) = 0 on∂D, (1.2) and a typical quantity of interestE[u^ε(x,·)], which is traditionally approximated using a Monte Carlo method.

Introducing a set of M realizations of the stochastic matrix {A^ε,m}_1≤m≤M, a direct, na¨ıve application of the MsFEM paradigm would consist in first computing for each realization m the stochastic MsFEM basis functionsφ^ε,m_i (x, ω), next performing a Galerkin approximation of (1.2) using this MsFEM basis set to compute {u^m_M(x, ω)}_1≤m≤M, and eventually approximatingE[u^ε(x,·)] by

E[u^ε(x,·)]≈ 1 M

M m=1

u^m_M(x, ω).

Such an approach is unpractical because of the prohibitively expensive computational load.

To reduce the computational cost and make the MsFEM approach practical in such a stochastic context, a natural idea we investigate in this article is to consider a less generic setting, for which a dedicated, more computationally affordable approach, can be designed. One possibility is to consider matricesA^ε(x, ω)≡A^ε(x)+

B(x, ω) in (1.2) that are not highly oscillatory in their stochastic part. In such cases, dedicated approaches have been proposed, we refer to [35] for more details. Another approach is to reduce the number of Monte−Carlo

simulations used for the computation of the multiscale basis functions. In [1,26], the authors assume that their coefficient can be written as a Karhunen-Lo`eve type expansion, and apply a collocation method to a priori choose some sparse realizations for which they compute the multiscale basis functions.

In this article, we consider one of the many alternate variants of problem (1.2). We suppose that A^ε(x, ω) is highly oscillatory in both its deterministic and stochastic components, but that it is a perturbation of a deterministic matrix. More precisely, we assume that

A^ε(x, ω)≡A^ε_η(x, ω) =A^ε₀(x) +ηA^ε₁(x, ω), (1.3) whereA^ε₀ is a deterministic matrix andηis a small deterministic parameter. This model may be well suited for heterogeneous materials (or, more generally, media) that, although not periodic, arenot fullystochastic, in the sense that they may be considered as aperturbationof a deterministic material. We call this setting theweakly stochastic setting. Note that many practical situations, involving actual materials or media, can be considered, at a good level of approximation, as perturbations of a deterministic (often periodic) setting (seee.g.[41]).

In a series of recent works (see [14,15,25] and [6–8]; see also [5] for a unified presentation), we have considered such a setting, in the context of homogenization theory (the matrixA^ε_η(x, ω) in (1.2)−(1.3) readsA^ε_η(x, ω) = A_η(x/ε, ω) for a stationary matrixA_η(x, ω), which is, in a sense to be made precise, a perturbation of a periodic matrix). We have shown there that, in such a case, the workload for computing the homogenized solution is significantly lighter than for generic stochastic homogenization, and actually comparable to the workload for periodic homogenization. We will show in the sequel that the MsFEM can be adapted to this weakly stochastic setting, providing an approximation of the solutionu^ε_η to (1.2)−(1.3), for fixedε, at a much smaller computational cost than the direct approach.

The main idea of our proposed approach is to compute a set of deterministicMsFEM basis functions using A^ε₀, the deterministic part of A^ε_η in the expansion (1.3), and then to perform Monte Carlo realizations at the macroscale level using a set ofMrealizations of the random matrix

A^ε,m_η (x, ω)

1≤m≤M (see Sect.2 for a detailed presentation). Note that, for each of these realizations, we solve theoriginalproblem, with thecomplete matrixA^ε_η, and not only its deterministic part. Only the basis set is taken deterministic. By construction, the approach provides an approximation

u_S(x, ω) = L i=1

(U_S(ω))_i φ^ε_i(x)

ofu^ε_η(x, ω), where the basis functionsφ^ε_i aredeterministic. These basis functions are computed only once, hence the cost to compute {u^m_S(x, ω)}_1≤m≤M is much smaller than the cost to compute {u^m_M(x, ω)}_1≤m≤M. This is especially true if (1.2) has to be solved for many right-hand sidesf. We expect that this approximationu_S is as accurate asu_M for smallη. We show below that this is indeed the case, whenA^ε_η is a perturbation ofA^ε₀(see Sect.3for numerical tests).

We would like to note that the MsFEM is not the only multiscale technique based on finite elements. The bottom line of our approach, consisting of generating suitable multiscale functions for the discretization of a weakly stochastic problem, using for this purpose the deterministic reference problem, can in principle be applied to other multiscale techniques. Another popular technique is the HMM method [27–29], for which our approach could in principle be easily adapted.

In the numerical tests reported on in Section3, we compare, in theH¹ norm,u^ε_η (the exact solution to (1.2) with the matrixA^ε≡A^ε_η given by (1.3)) withu_S (the solution provided by our approach) andu_M (the solution provided by the ideal, expensive approach). The quantityu^ε_η−uM_H¹_(D)represents the best possible accuracy that we can achieve, in the sense that our approach inherits the limitations of the MsFEM approach. We thus cannot expect our approximation u_S to be more accurate than u_M. We can only hope to compute an approximation of comparable quality with a much reduced workload. The numerical results we obtain confirm that, for small η in (1.3), the quantityuS−u^ε_ηH¹_(D) is of the same order of magnitude asuM−u^ε_ηH¹_(D),

ET AL.

although, we repeat it, the computational cost to compute u_S is much smaller than that to computeu_M. In Section3, we also show the advantage of performing Monte Carlo realizations at the macroscale level (using the random matrixA^ε,m_η ) over solving a deterministic macroscale problem with only the deterministic part ofA^ε_η.

We next derive error bounds for our approach in Section 4. We recall that, in the deterministic setting, a classical context for proving convergence of the MsFEM approach is the case when, in the reference problem (1.1), the matrix reads A^ε(x) = A_per

x ε

for a fixed periodic matrix A_per. Likewise, to be able to perform our theoretical analysis in the stochastic setting, we assume in Section 4 that A^ε_η(x, ω) = A_η

x ε, ω

for a fixed stationary random matrixA_η. The problem (1.2)−(1.3) then admits a homogenized limit whenεvanishes.

Our proof follows the same lines as that in the deterministic setting, which we now briefly review (see the introduction of Sect.4for more details on the structure of the proof). The MsFEM is a Galerkin approximation, that we assume momentarily, for the sake of clarity, to be aconformingapproximation (this is indeed the case when, for defining the highly oscillatory basis functionsφ^ε_i, oversampling isnotused). The error is then estimated using the C´ea lemma:

u^ε−u_MH¹_(D)≤C inf

vh∈Wh

u^ε−v_hH¹_(D),

where u^ε is the solution to the reference deterministic highly oscillatory problem (1.1), u_M is the MsFEM solution and C is a constant independent of ε and h. Taking advantage of the homogenization setting, we introduce the two-scale expansion

v^ε=u+ε d i=1

w⁰_e

i

· ε

∂_iu

ofu^ε, whereuis the homogenized solution,w⁰_ei is the periodic corrector associated toe_i∈R^d, and∂_iudenotes the partial derivative ∂u

∂x_i. We next write u^ε−u_MH¹_(D)≤C

u^ε−v^ε_H¹_(D)+ inf

vh∈Wh

v^ε−v_hH¹_(D)

.

The first term in the above right-hand side is estimated using standard homogenization results on therate of convergence of v^ε−u^ε. To estimate the second term, one considers a suitably chosen element v_h ∈ Wh, for whichv^ε−v_hH¹_(D)can be directly bounded.

Following the same strategy in our stochastic setting, we estimate the distance between the solutionu^ε_η to the reference stochastic problem (1.2)−(1.3) and the weakly stochastic MsFEM solutionu_S as

u^ε_η(·, ω)−u_S(·, ω)H¹(D)≤C

u^ε_η(·, ω)−v^ε_η(·, ω)H¹(D)+ inf

vh∈Wh

v_η^ε(·, ω)−v_hH¹(D)

.

We observe that a key ingredient for the proof is the rate of convergence of the difference between the reference solution u^ε_η and its two-scale expansion v^ε_η. Such a result is classical in periodic homogenization, but, to the best of our knowledge, open in the general stationary case (in dimensions higher than one). One only knows thatu^ε_η−v_η^εvanishes (in some appropriate norm) whenε→0. However, in the particular case whenA^ε_ηis only weaklystochastic, it is possible to obtain such a result, as we have shown in [42]. Hence, exploiting the specificity of our weakly stochastic setting, we are able to obtain (see our main result, Thm.4.5and estimate (4.37)):

E

u^ε_η−u_S²_H1 h

≤C √

ε+h+ε h+η

ε h

_d/2

ln(N(h)) +η+η²C(η)

, where · _H¹

h is a brokenH¹norm,C is a constant independent ofε,handη,Cis a bounded function asηgoes to 0, and N(h) is the number of elements in the mesh (roughly of order h^−d in dimensiond). As is often the

case in the deterministic setting, we use here (both for our numerical tests and in the analysis) the oversampling technique. Consequently, the basis functions φ^ε_i do not belong to H₀¹(D), hence the use of a brokenH¹ norm in the above estimate. As we point out below, when η = 0 in (1.3), our approach reduces to the standard deterministic MsFEM (with oversampling), and the above estimates then agree with those proved in [32].

This article is organized as follows. First, in Section 2, we describe the MsFEM approach. For consistency, we begin by the deterministic setting in Section 2.1, and point out there that the direct adaptation to the general stochastic setting yields a prohibitively expensive approach. The adaptation of the approach to the weakly stochastic settingis described in Section 2.2. We next turn to numerical simulations, in Section3. Some procedures to efficiently implement the approach are first described in Section 3.1. We next consider a one- dimensional test (see Sect.3.2), which is useful for several reasons. First, it allows to calibrate some numerical parameters, such as the number M of independent realizations when estimating the exact expectation by an empirical mean. Second, we assess the accuracy of our approach with respect to the magnitude of η. We demonstrate there thatη does not have to be extremely small for our method to be very efficient. For instance, on the test case considered in Section 3.2, we show that our approach is as accurate as the expensive, direct approach as soon asη is such that

ηa^ε₁ a^ε₀

L^∞(R×Ω)

is equal to or smaller than 0.1,

wherea^ε₀is the deterministic component of the diffusion coefficienta^ε_η andηa^ε₁ is the stochastic component (see expansion (1.3)). On the other hand, as pointed out above, our approach is not meant to address the regime whenη ≈1. Lastly, we also assess the accuracy of our approach with respect to the presence of frequencies in the random coefficient a^ε_η that are not taken into account in the MsFEM basis set. We next turn to two test cases in dimension two, where we observe that our approach performs as well as in the one-dimensional case (see Sect. 3.3). In particular, in Section 3.3.2, we successfully address a classical test-case of the literature. In Section3.4, we compare our approach with a fully deterministic approach. All the information about variance is lost when using the latter approach. In contrast, using our approach, we show that we can accurately approximate some quantities of interest which are random in nature, such as the variance of the solution.

Section 4 is devoted to the analysis of the approach, in the homogenization setting (i.e. when the matrix in (1.2) reads A^ε(x, ω) = A

x ε, ω

where A is stationary). Our main result, Theorem 4.5, is presented in Section 4.1, and proved in Section 4.2. The proofs of some technical results are collected in Appendix A. In addition, we specifically consider the one dimensional case in Section4.3.

2. MsFEM-type approaches

For consistency and the convenience of the reader, we present in this section the MsFEM approach to solve the original elliptic problem (1.1). For clarity, we begin by presenting the approach in a deterministic setting.

The reader familiar with the MsFEM may easily skip this section and directly proceed to Section2.2, where we present our approach in a weakly stochastic setting.

2.1. Description in a classical deterministic setting

Letu^ε∈H¹(D) be the solution to (1.1), where the matrixA^ε∈(L^∞(D))^d×d satisfies the standard coercivity condition: there exists two constantsa₊≥a₋ >0 such that, for anyε,

∀ξ∈R^d, a₋|ξ|²≤ξ^TA^ε(x)ξa.e. inD and A^ε_L^∞_(D)≤a₊.

Note that the MsFEM approach is not restricted to the periodic setting. We therefore do not assume that A^ε(x) =A_per(x/ε) for a fixed periodic matrix A_per.

ET AL.

The MsFEM approach consists in performing a variational approximation of (1.1) where the basis functions are precomputed and encode the fast oscillations present in (1.1). In the sequel we argue on the following formulation, equivalent to (1.1):

Findu^ε∈H₀¹(D) such that, for any v∈H₀¹(D), Aε(u^ε, v) =b(v), (2.1) where

Aε(u, v) =

D(∇v(x))^TA^ε(x)∇u(x) dx and b(v) =

Df(x)v(x) dx.

The MsFEM is a three-step approach:

1. introduce a standard discretization of the domain D using a coarse mesh as compared to the small scale oscillations ofA^ε;

2. for each elementKof the coarse mesh, compute the basis functionφ^ε,K_i as the solution of an elliptic equation posed inK(seee.g.(2.7)−(2.8)−(2.9) below);

3. solve the Galerkin approximation of (2.1), for the set of basis functions defined at Step 2.

The advantage of the approach is that, for the same accuracy of the approximation as that provided by a standard FEM, the macroscale mesh can be chosen sufficiently coarse so that the resulting discretized problem has a limited number of degrees of freedom, and may thus be computationally solved inexpensively. This is observed in practice [37], and proven by a theoretical analysis (see [32,38]) when the problem (2.1) admits a homogenized limit. See also [30] and references therein.

To further illustrate this fact, we reproduce here a simple one-dimensional analysis we borrow from A. Lozinski (see [44], Chap. 6 and [17]). This analysis explains remarkably well the interest of the approach, and, in contrast to [32,38], is not restricted to a homogenization setting. Consider the one-dimensional domainD= (0,1) and the reference problem

Lu=f, u(0) =u(1) = 0,

for the operatorLu:=−(νu), wheref ∈L²(0,1) andν∈L^∞(0,1) withν(x)≥ν_min>0 almost everywhere on (0,1). The functionν may have oscillations at a small scale. The associated weak formulation reads

Find u∈H₀¹(0,1) such that, for anyv∈H₀¹(0,1), a(u, v) =b(v), (2.2) with

a(u, v) = ₁

0

ν(x)u(x)v(x) dx and b(v) = ₁

0

f(x)v(x) dx.

We now introduce the nodes 0 = x₀ < x₁ < · · · < x_L = 1 that define the elements K_i = [x_i−1, x_i]. Let h= max|xi−x_i−1|be the mesh size. The multiscale finite element space

Wh=

v_h∈C⁰(0,1) such that Lvh= 0 on eachK_i

, (2.3)

defined using the operatorL, isadaptedto the problem under study. We next proceed with a Galerkin approx- imation of (2.2) using the spaceWh:

Find u_h∈ Wh such that, for any v_h∈ Wh, a(u_h, v_h) =b(v_h).

The solutionu_h then satisfies

u−u_hE≤ h π√

ν_minf_L²_(0,1) (2.4)

where · E =

a(·,·) is the energy norm. The proof of this estimate goes as follows. By definition of uand u_h, we havea(u−u_h, v_h) = 0 for anyv_h ∈ Wh. Hence,u_his the orthogonal projection ofuonWh according to the scalar producta(·,·). Since · E is the norm associated to that scalar product, we have

u−u_hE= inf

vh∈Wh

u−v_hE. (2.5)

Choosev_h to be the finite element interpolant ofu, which is defined by v_h(x_i) =u(x_i) for anyi= 0,1, . . . , L, and consider the interpolation errore=u−v_h. On each elementK_i, we have, precisely because the spaceW_h is defined as (2.3),

Le=−(νe)=f with e(x_i−1) =e(x_i) = 0.

We multiply bye, integrate by part and obtain _xi

xi−1

ν(x)|e(x)|²dx= _xi

xi−1

f(x)e(x) dx≤ f_L²_(Ki)e_L²_(Ki). (2.6) Sinceevanishes on the boundary ofK_i, the Poincar´e inequality with the best constant (x_i−x_i−1)/πyields

e_L²_(Ki)≤x_i−x_i−1

π e_L²_(Ki)≤ h π√

ν_min xi

xi−1

ν(x)|e(x)|²dx 1/2

. By substitution in (2.6), we obtain

_xi

xi−1

ν(x)|e(x)|²dx≤ h²

π²ν_minf²_L2(Ki).

Summing over the elements and using (2.5) yields (2.4). Using again thatν is bounded from below, we deduce from (2.4) that

u−u_hH¹(0,1)≤ h

C_Dπ ν_minfL²(0,1),

whereC_Dis the Poincar´e constant of the domainD= (0,1). As pointed out in Chapter 6 of [44], the interest of the above estimate (or of estimate (2.4)) lies in the fact that the constant in the right-hand side only depends onν throughν_min, and remains the same even ifν oscillates at a small scale. In contrast, for a standard finite element method, the error is also proportional toh, but with a constant that depends on theH² norm of the exact solutionu. With a standard finite element spaceWh, we indeed classically deduce by C´ea’s lemma that

u−u_hH¹_(0,1)≤ νL^∞(0,1)

C_Dν_min inf

vh∈Wh

u−v_hH¹_(0,1)= νL^∞(0,1)

C_Dν_min u−R_hu_H¹_(0,1),

where C_D is the Poincar´e constant of the domainD= (0,1), andR_huis the projection ofuonWh according to theH¹ scalar product. We thus obtain that

u−u_hH¹(0,1)≤ChνL^∞(0,1)

ν_min uL²(0,1),

where C is independent from the functionsν andu. If ν oscillates at a small scale (e.g. ν(x) = ν(x/ε) for a fixed functionν), theH²norm ofumay be large (of the order ofε⁻¹). A FEM approach then requireshto be smaller thanεto reach a good accuracy.

We conclude this illustration by noting that such a general analysis of the MsFEM approach is not available in dimensiond≥2. The analysis presented in [32,38], which is performed without any restriction on the dimension, additionally assumes that the matrixA^ε in (2.1) readsA^ε(x) =A_per(x/ε) for a fixedperiodic matrixA_per.

We now describe the MsFEM in a multidimensional setting.

2.1.1. Definition of the coarse mesh

For simplicity (see Rem.2.1below), we consider a classicalP1 discretization of the domainD. We denote by Th the corresponding mesh, withLnodes. Let φ⁰_i, i= 1, . . . , L, be the basis functions. We introduce the finite element space

Vh:= span(φ⁰_i, i= 1, . . . , L),

ET AL.

Figure 1. Definition ofS(in 2D for clarity).

and define the restriction

φ^0,K_i :=φ⁰_i

K

of these functions in each elementK.

Remark 2.1. We refer to [3] for a presentation of a MsFEM method that usesP2 macroscale basis functions.

2.1.2. Definition of the MsFEM basis

Several definitions of the MsFEM basis functions have been proposed in the literature (see e.g. [3,30,32,37–39]). They all follow the same pattern but they give rise to various methods. We present in the following the particular method that we have implemented. It makes use of the oversampling technique introduced in [37] and developed in [36].

For any elementK, we consider a domainS⊃K (see Fig.1), obtained from K by an homothetic transfor- mation of center the centroid ofK, and of ratio larger than 1.

Letx^S_j denote the coordinate of the vertexj of the domainS. For any vertexiofS, we introduce the affine functionχ^0,S_i (defined onS) that satisfies the conditionχ^0,S_i (x^S_j) =δ_ij for allj. Letχ^ε,S_i ∈H¹(S) be the unique solution to the problem

−div

A^ε(x)∇χ^ε,S_i (x)

= 0 inS, χ^ε,S_i =χ^0,S_i on∂S, (2.7) which, in practice, is numerically solved e.g. using a finite element method with a mesh size adapted to the small scaleε. We then define the local basis functions

φ^ε,K_i = d+1 j=1

α_ij χ^ε,S_j

K (2.8)

as linear combinations of the restrictions ofχ^ε,S_i onK, withα_ij chosen such that

∀1≤i, j≤d+ 1, φ^0,K_i (x^K_j ) =

d+1

j=1

α_ijχ^0,S_j (x^K_j ) =δ_ij, (2.9)

wherex^K_j denotes the coordinate of thejth vertex of the elementK. Note that the condition (2.9) is enforced on the function φ^0,K_i , and not onφ^ε,K_i . The coefficientsα_ij are consequently independent fromε. Asφ^0,K_i and

χ^0,S_j

K are both affine onK, condition (2.9) implies that

∀1≤i≤d+ 1, ∀x∈K, φ^0,K_i (x) =

d+1

j=1

α_ijχ^0,S_j (x). (2.10)

We next introduce the functionsφ^ε_i defined on Dby φ^ε_i|_K=φ^ε,K_i for all elementsK.

Note that the problems (2.7), indexed by S, are all independent from one another. They may be solved in parallel.

2.1.3. Macroscale problem

We now introduce the finite dimensional space

Wh:= span(φ^ε_i, i= 1, . . . , L), and proceed with the approximation

Find u_M ∈ Wh such that, for any v∈ Wh, A^h_ε(u_M, v) =b(v), (2.11) of (2.1), where

A^h_ε(u, v) =

K

K(∇v(x))^TA^ε(x)∇u(x) dx and b(v) =

Df(x)v(x) dx.

Observe thatφ^ε_i has jumps across the edges of the triangulation (due to the use of the oversampling technique), henceWh ⊂H¹(D), thus the broken integral used to defineA^h_ε(u, v). On the other hand, sinceWh ⊂L²(D), the linear formbis well defined forv∈ Wh. The formulation (2.11) is anon-conformingGalerkin approximation of (2.1). This brings additional error terms in the error estimation (see Lem.4.7in Section4). On another note, remark that the dimension of Wh is equal to L. The formulation (2.11) hence requires solving a linear system with only a limited number of degrees of freedom.

We are now in position to substantiate our claim in the introduction, where we briefly mentioned that, in the stochastic setting, a direct application of the MsFEM to approximate the solution to (1.2) isunpractical. It would indeed lead to compute, foreach realizationofA^ε(x, ω), first a basis set and second a macroscale solution.

This approach has been briefly examined theoretically in [21]. It is prohibitively expensive. We therefore turn to an alternate approach.

2.2. A weakly stochastic setting

We now restrict the general setting and propose a dedicated, practical MsFEM type approach. Following up on previous works (see [5,13,24,41]) and as announced in (1.3), we assume here that the random matrix A^ε(x, ω) in (1.2) is aperturbation of a deterministic matrix, in the sense that

A^ε(x, ω)≡A^ε_η(x, ω) =A^ε₀(x) +η A^ε₁(x, ω), (2.12) whereη∈Ris a small deterministic parameter,A^ε₀andA^ε₁are bounded matrices, andA^ε₀is coercive, uniformly inε. We also assume that the matrixA^ε_η itself satisfies the coercivity and boundedness assumptions, uniformly in ηand ε(we refer to [6–8] and [15,25] for other perturbative settings).

The principle of the proposed approach is to compute the MsFEM basis set of functions with the deter- ministic part A^ε₀ of the matrix A^ε_η, and next to perform Monte−Carlo realizations for the macroscale prob- lem (1.2)−(2.12), where we keep the exact matrix A^ε_η (and not only its deterministic part). Following the approach sketched in Section 2.1, we first solve (2.7) with A^ε(x) ≡A^ε₀(x), and build the deterministic finite dimensional space

Wh:= span(φ^ε_i, i= 1, . . . , L)

ET AL.

following (2.8)−(2.9). We next proceed with a standard Galerkin approximation of (1.2)−(2.12) usingWh. For eachm∈ {1, . . . , M}, we consider a realizationA^ε,m_η (·, ω) and computeu^m_S(·, ω)∈ W_hsuch that

∀v∈ Wh,

K

K(∇v(x))^TA^ε,m_η (x, ω)∇u^m_S(x, ω) dx=

Df(x)v(x) dx. (2.13) Since the MsFEM basis functions are only computed once (rather than for each realization ofA^ε_η(x, ω)), a large computational gain is expected, and obtained, in comparison to the direct approach described above.

3. Numerical simulations

This section is devoted to the many numerical simulations we have performed. We first discuss some imple- mentation details. Next, we numerically estimate the performance of our approach on various test cases, and assess its sensitivity with respect to the magnitude of η. We consider in Section3.2 a test case in dimension one. In Section3.3, we next study two test cases in dimension two. We also study how the presence inA^ε₁(the random component of the matrixA^ε_η) of high frequencies that are not present in the deterministic component A^ε₀, and that are thus not encoded in the highly oscillatory basis functions, affects the accuracy of our approach.

In Section3.4, we eventually compare our approach with a fully deterministic approach.

Let u^ε_η be the reference solution to (1.2)−(1.3) obtained using a finite element method with a mesh size adapted to the small scale ε,u_S be the approximation given by our approach (described in Sect.2.2) andu_M be the approximation given by the direct approach (in which the MsFEM basis set is recomputed for each realization A^ε,m_η (x, ω), as explained at the end of Sect. 2.1). Our goal is to compare the error u_S −u^ε_η of our numerical approximation with the erroru_M −u^ε_η of the direct and expensive approach. When η is small, we expect the approximationu_S to be essentially as accurate as the approximationu_M, and we show below that this is indeed the case.

In the sequel, we assess the accuracy using the estimators e_L2(u₁, u₂) =E

u1−u₂L²(D)

u2_L²_(D)

and e_H1(u₁, u₂) =E

u1−u_2H¹ u2_H¹ h

h

, (3.1)

whereu₁ andu₂ are the solutions obtained with any two different methods, and u_H¹

h:=

K∈Th

u²_H1(K)

_1/2

(3.2) is the broken H¹ norm. The expectation is in turn computed using a Monte−Carlo method. Considering M realizations {Xm(ω)}_1≤m≤M of a random variable, e.g. X(ω) = u1(·, ω)−u₂(·, ω)_H¹

u2(·, ω)_H¹ h h

, we compute the empirical meanμ_M and the empirical standard deviationσ_M as

μ_M(X) = 1 M

M m=1

X_m(ω), σ_M² (X) = 1 M −1

M m=1

(X_m(ω)−μ_M(X))². (3.3) As a classical consequence of the Central Limit Theorem, the following estimate is commonly employed:

|E(X)−μ_M(X)| ≤1.96 σ_M(X)

√M ·

It provides a practical evaluation ofE(X) from the knowledge ofμ_M(X) andσ_M(X). The numerical parameters have been determined by an empirical study of convergence. For instance, for the reference solution, we choose

the mesh sizehsuch that the quantity u^ε,h_η −u^ε,h/2_η H¹(D)

u^ε,h/2η _H¹_(D) is smaller than 0.03%, thereby formally admitting that the approximation has converged inh. The MsFEM parameters are determined likewise.

All the computations have been performed using FreeFem++ [33], with the MPI tools.

3.1. Implementation details

In the deterministic version of the MsFEM, the same matrix A^εappears in the definition (2.7) of the basis functions and in the macroscale variational formulation (2.11). This can be used to expedite the computation of the stiffness matrix associated with (2.11). In our approach, described in Section2.2, the matrix that appears in the definition of the basis functions isA^ε₀, whereas the macroscale variational problem involvesA^ε_η≡A^ε₀+ηA^ε₁. An additional numerical computation is thus needed.

To solve (2.13), we need to compute, for each element Kand each realizationA^ε,m_η (x, ω), the integrals K^η,m_ij (ω) =

K

∇φ^ε,K_i (x) _T

A^ε,m_η (x, ω)∇φ^ε,K_j (x) dx, (3.4) where φ^ε,K_i are deterministic functions. We recall that A^ε_η(x, ω) =A^ε₀(x) +ηA^ε₁(x, ω) (see (2.12)). To allow for an efficient evaluation of (3.4), we assume henceforth thatA^ε₁is of the form

A^ε₁(x, ω) =

k∈Z^d

1_Q+k x

ε

X_k(ω)B^k_ε(x), (3.5)

where Q= (0,1)^d, where (X_k)_k∈Zd are scalar random variables, and for anyk ∈Z^d, x→B^k_ε(x)∈ R^d×d are some deterministic functions. We comment on this assumption in Remark3.1below. The important consequence of (3.5) is that we can write the integral (3.4) as a linear combination ofdeterministic integrals over cells of size ε, withrandom coefficients. To simplify the notation, we assume that the spatial dimension isd= 2. We define

p=

min y_i

ε,y_j ε,y_k

ε

, q=

max y_i

ε,y_j ε,y_k

ε

+ 1,

where y_i, y_j and y_k are the y-axis coordinates of the three vertex of K (see Fig. 2). We likewise define the integersl andm(see Fig. 2). We can then write (3.4) as

K_ij^η,m(ω) =

K

∇φ^ε,K_i (x) _T

A^ε,m_η (x, ω)∇φ^ε,K_j (x) dx=K^0,K_ij +η

q−1

α=p m−1

β=l

X_α,β^m (ω)K^1,K_αβij, (3.6) where

K^0,K_ij =

K

∇φ^ε,K_i (x) _T

A^ε₀(x)∇φ^ε,K_j (x) dx, (3.7)

K^1,K_αβij =

(α+1)ε

αε

(β+1)ε

βε

1_K(x)

∇φ^ε,K_i (x) _T

B^α,β_ε (x)∇φ^ε,K_j (x) dx. (3.8)

We thus compute once the deterministic integrals (3.7) and (3.8). Next, for each realization ofA^ε_η, we evaluate the stiffness matrix elementsK_ij^η,m(ω) using the right hand side of (3.6). No numerical quadrature is needed. As a consequence of (3.5), most of the work for assembling the stiffness matrix is only performed once, independently of the number of Monte Carlo realizations. This significantly contributes to the gain in term of computational cost.

ET AL.

Figure 2. To practically compute the integral (3.4), we write that each elementK (here in dimensiond= 2) is a subset of a quadrangle (here [lε, mε]×[pε, qε]) composed of cells of sizeε^d.

Remark 3.1. Assumption (3.5) is quite general, and already covers many interesting cases in practice. As explained above, the point in (3.5) is thatA^ε₁ is adirect product(or here, a sum of direct products) of a function depending on xwith a random variable that only depends onω. Otherwise stated, A^ε₁(x, ω) depends linearly, in an explicit way, ofω. A similar assumption is made when applying reduced basis methods [45] to a problem of the type

Find u_λ such that, for anyv,a(u_λ, v;λ) =b(v), (3.9) where a(·,·;λ) is a bilinear form parameterized by λ. Assume this problem has been solved for some values {λi}^I_i=1of the parameter, yielding the functions{uλi}^I_i=1. Under the assumption thata(·,·;λ) =a₀(·,·)+λa1(·,·) (namely, a(·,·;λ) depends linearly on λ), one can precompute the stiffness matrix elements a₀(u_λi, u_λj) and a₁(u_λi, u_λj) for any 1 ≤ i, j ≤ I. This allows to next perform a very efficient Galerkin approximation of the problem (3.9) (for anyλ) on the space Span(u_λi, i= 1, . . . , I).

3.2. One-dimensional test-case

The purpose of this section is threefold. We first calibrate the numberM of realizations considered for the Monte−Carlo method for the two-dimensional numerical experiments that we consider in the sequel. We next investigate how the accuracy of our approach depends onη and on the presence of frequencies in the random coefficienta^ε_η that are not taken into account in the MsFEM basis set functions. The low computational costs that we face in this one-dimensional situation allow us to test our approach more comprehensively than in the two-dimensional test-cases described below.

Let (X_k)_k∈Z denote a sequence of independent, identically distributed scalar random variables uniformly distributed in [0,1]. We consider the random coefficient

a^ε_η(x, ω) =

k∈Z

1_(k,k+1]

x

ε 5 + 50 sin² πx

ε

+ηX_k(ω)κ sin² ζπx

ε

,

0 5 10 15 20 25 30 35 40 45 50

−2 0 2 4 6 8 10 12 14 16x 10⁻⁴

M

Figure 3. Convergence of the indicatore_H1(u_M, u^ε_η) (see (3.1)), forη= 1,ζ= 1 andκ= 55.

For each value ofM, we plot the empirical mean along with its confidence interval, computed from the firstM realizations. We only plot the results for the first 50 realizations.

which is a particular example of the expansion (2.12) with a^ε₀(x) = 5 + 50 sin²

πx ε

and a^ε₁(x, ω) =

k∈Z

1_(k,k+1]

x ε

X_k(ω)κsin² ζπx

ε

, and that satisfies the structural assumption (3.5). We setε= 0.025 and chooseκsuch that the quantity

R(κ, ζ) = a^ε₁

a^ε₀

L^∞(D×Ω)

= SupEss_ω∈Ω a^ε₁(·, ω)

a^ε₀

L^∞(D)

(3.10) has the same valueR(κ, ζ) = 1 for the three different values ofζ={1,3,7}we consider below. This yields the choices (κ, ζ) = (55,1), (κ, ζ) = (14.38,3) and (κ, ζ) = (8.39,7). We analytically compute the reference function u^ε_η, solution to

− d dx

a^ε_η(x, ω)du^ε_η dx(x, ω)

= 1 in (0,1), u^ε_η(0, ω) =u^ε_η(1, ω) = 0,

as well as the MsFEM basis functions for both approaches. Letu_M andu_S be the approximation ofu^ε_η by the two MsFEM approaches described above, where the coarse mesh size ish= 1/30.

We first calibrate the number of independent realizations to accurately approximate the exact expectation in (3.1) by the empirical mean (3.3). To this aim, we present on Figure3 the mean and the confidence interval computed using (3.3) for an increasing numberM of realizations (we compute up to 1000 independent realiza- tions). We check that this indicator reaches a plateau forM ≥30, and thus converges fast. On this example, considering 30 realizations is hence sufficient to accurately compute the error (3.1). Based on this observation, we will only considerM = 30 realizations in the two dimensional examples of Section3.3.

Remark 3.2. There is no reason to think that the calibration of our parameters that we perform in the one- dimensional situation provides an adequate adaptation of these parameters for the higher dimensional setting.